Georgia Institute of Technology College of Sciences

Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. We introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems. This talk is based on joint work with Gagan Goel, Chinmay Karande and Wang Lei. This is a joint ACO/DOS seminar, so please come a little early for pizza and refreshments sponsored by ACO.

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; price equilibria in markets; optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysis of the evolution of various types of dynamic stochastic models. It is not known whether these problems can be solved in polynomial time. Despite their broad diversity, there are certain common computational principles that underlie different types of equilibria and connect many of these problems to each other. In this talk we will discuss some of these common principles and the corresponding complexity classes that capture them; the effect on the complexity of the underlying computational framework; and the relationship with other open questions in computation.

Concentration results for the TSP, MWST and many other problems with random inputs show the answer is concentrated tightly around the mean. But most results assume uniform density of the input. We will generalize these to heavy-tailed inputs which seem to be ubiquitous in modern applications. To accomplish this, we prove two new general probability inequalities. The simpler first inequality weakens both hypotheses in Hoffding-Azuma inequality and is enough to tackle TSP, MWST and Random Projections. The second inequality further weakens the moment requirements and using it, we prove the best possible concentration for the long-studied bin packing problem as well as some others. Many other applications seem possible..

We will briefly review the definition of the Cech cohomology groups of a sheaf (so if you missed last weeks talk, you should still be able to follow this weeks), discuss some basic properties of the Cech construction and give some computations that shows how the theory connects to other things (like ordinary cohomology and line bundles).